Open Access

Jost Solution and the Spectrum of the Discrete Dirac Systems

Boundary Value Problems20102010:306571

DOI: 10.1155/2010/306571

Received: 14 September 2010

Accepted: 10 November 2010

Published: 29 November 2010

Abstract

We find polynomial-type Jost solution of the self-adjoint discrete Dirac systems. Then we investigate analytical properties and asymptotic behaviour of the Jost solution. Using the Weyl compact perturbation theorem, we prove that discrete Dirac system has the continuous spectrum filling the segment [-2,2]. We also study the eigenvalues of the Dirac system. In particular, we prove that the Dirac system has a finite number of simple real eigenvalues.

1. Introduction

Let us consider the boundary value problem (BVP) generated by the Sturm-Liouville equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ1_HTML.gif
(1.1)
and the boundary condition
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ2_HTML.gif
(1.2)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq1_HTML.gif is a real-valued function and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq2_HTML.gif is a spectral parameter. The bounded solution of (1.1) satisfying the condition
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ3_HTML.gif
(1.3)
will be denoted by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq3_HTML.gif . The solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq4_HTML.gif satisfies the integral equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ4_HTML.gif
(1.4)
It has been shown that, under the condition
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ5_HTML.gif
(1.5)
the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq5_HTML.gif has the integral representation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ6_HTML.gif
(1.6)
where the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq6_HTML.gif is defined by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq7_HTML.gif . The function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq8_HTML.gif is analytic with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq9_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq10_HTML.gif , continuous https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq11_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ7_HTML.gif
(1.7)

holds [1, chapter 3].

The functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq12_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq13_HTML.gif are called Jost solution and Jost function of the BVP (1.1) and (1.2), respectively. These functions play an important role in the solution of inverse problems of the quantum scattering theory [14]. In particular, the scattering date of the BVP (1.1) and (1.2) is defined in terms of Jost solution and Jost function. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq14_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq15_HTML.gif , be the zeros of the Jost function, numbered in the order of increase of their moduli ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq16_HTML.gif ) and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ8_HTML.gif
(1.8)
The functions
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ9_HTML.gif
(1.9)
are bounded solutions of the BVP (1.1) and (1.2), where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq17_HTML.gif is the scattering function [14]. Using (1.7), we get that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ10_HTML.gif
(1.10)

hold. The collection of quantities https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq18_HTML.gif that specify to as the behaviour of the radial wave functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq19_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq20_HTML.gif at infinity is called the scattering of the BVP (1.1) and (1.2).

Let us consider the self-adjoint system of differential equations of first order
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ11_HTML.gif
(1.11)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq21_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq22_HTML.gif are real-valued continuous functions. In the case https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq23_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq24_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq25_HTML.gif is a potential function and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq26_HTML.gif the mass of a particle, (1.11) is called stationary Dirac system in relativistic quantum theory [5, chapter 7]. Jost solution and the scattering theory of (1.11) have been investigated in [6].

Jost solutions of quadratic pencil of Schrödinger, Klein-Gordon, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq27_HTML.gif -Sturm-Liouville equations have been obtained in [79]. In [1017], using the analytical properties of Jost functions, the spectral analysis of differential and difference equations has been investigated.

Discrete boundary value problems have been intensively studied in the last decade. The modelling of certain linear and nonlinear problems from economics, optimal control theory, and other areas of study has led to the rapid development of the theory of difference equations. Also the spectral analysis of the difference equations has been treated by various authors in connection with the classical moment problem (see the monographs of Agarwal [18], Agarwal and Wong [19], Kelley and Peterson [20], and the references therein). The spectral theory of the difference equations has also been applied to the solution of classes of nonlinear discrete Korteveg-de Vries equations and Toda lattices [21].

Now let us consider the discrete Dirac system
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ12_HTML.gif
(1.12)
with the boundary condition
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ13_HTML.gif
(1.13)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq28_HTML.gif is the forward difference operator: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq29_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq30_HTML.gif is the backward difference operator: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq31_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq32_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq33_HTML.gif are real sequences. It is evident that (1.12) is the discrete analogy of (1.11). Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq34_HTML.gif denote the operator generated in the Hilbert space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq35_HTML.gif by the BVP (1.12) and (1.13). The operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq36_HTML.gif is self-adjoint, that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq37_HTML.gif . In the following, we will assume that, the real sequences https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq38_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq39_HTML.gif satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ14_HTML.gif
(1.14)

In this paper, we find Jost solution of (1.12) and investigate analytical properties and asymptotic behaviour of the Jost solution. We also show that, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq40_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq41_HTML.gif denotes the continuous spectrum of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq42_HTML.gif , generated in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq43_HTML.gif by (1.12) and (1.13).

We also prove that under the condition (1.14) the operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq44_HTML.gif has a finite number of simple real eigenvalues.

2. Jost Solution of (1.12)

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq45_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq46_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq47_HTML.gif from (1.12), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ15_HTML.gif
(2.1)
It is clear that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ16_HTML.gif
(2.2)
is a solution of (2.1). Now we find the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq48_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq49_HTML.gif of (1.12) for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq50_HTML.gif , satisfying the condition
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ17_HTML.gif
(2.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq51_HTML.gif .

Theorem 2.1.

Under the condition (1.14) for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq52_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq53_HTML.gif ,  (1.12) has the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq54_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq55_HTML.gif , having the representation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ18_HTML.gif
(2.4)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ19_HTML.gif
(2.5)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ20_HTML.gif
(2.6)

Proof.

Substituting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq56_HTML.gif defined by (2.4) and (2.5) into (1.12) and taking https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq57_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq58_HTML.gif , we get the following:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ21_HTML.gif
(2.7)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ22_HTML.gif
(2.8)
Using (2.7) and (2.8),
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ23_HTML.gif
(2.9)
hold, where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq59_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq60_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ24_HTML.gif
(2.10)

By the condition (1.14), the series in the definition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq61_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq62_HTML.gif ) are absolutely convergent. Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq63_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq64_HTML.gif ) can, by uniquely be defined by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq65_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq66_HTML.gif , that is, the system (1.12) for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq67_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq68_HTML.gif , has the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq69_HTML.gif given by (2.4) and (2.5).

By induction, we easily obtain that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ25_HTML.gif
(2.11)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq70_HTML.gif is the integer part of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq71_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq72_HTML.gif is a constant. It follows from (2.4) and (2.11) that (2.3) holds.

Theorem 2.2.

The solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq73_HTML.gif has an analytic continuation from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq74_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq75_HTML.gif .

Proof.

From (1.14) and (2.11), we obtain that the series https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq76_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq77_HTML.gif are uniformly convergent in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq78_HTML.gif . This shows that the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq79_HTML.gif has an analytic continuation from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq80_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq81_HTML.gif .

The functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq82_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq83_HTML.gif are called Jost solution and Jost function of the BVP (1.12) and (1.13), respectively. It follows from Theorem 2.2 that Jost solution and Jost function are analytic in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq84_HTML.gif and continuous on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq85_HTML.gif .

Theorem 2.3.

The following asymptotics hold:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ26_HTML.gif
(2.12)

Proof.

From (2.4), we get that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ27_HTML.gif
(2.13)
Using (2.11) and (2.13), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ28_HTML.gif
(2.14)
So we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ29_HTML.gif
(2.15)
by (2.14). In a manner similar to (2.15), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ30_HTML.gif
(2.16)

From (2.15) and (2.16), we obtain (2.12).

3. Continuous and Discrete Spectrum of the BVP (1.12) and (1.13)

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq86_HTML.gif denote the Hilbert space of all complex vector sequences
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ31_HTML.gif
(3.1)
with the norm
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ32_HTML.gif
(3.2)

Theorem 3.1.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq87_HTML.gif .

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq88_HTML.gif denote the operator generated in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq89_HTML.gif by the BVP
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ33_HTML.gif
(3.3)
We also define the operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq90_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq91_HTML.gif by the following:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ34_HTML.gif
(3.4)
It is clear that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq92_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ35_HTML.gif
(3.5)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq93_HTML.gif denotes the operator generated in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq94_HTML.gif by the BVP (1.12) and (1.13). It follows from (1.14) that the operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq95_HTML.gif is compact in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq96_HTML.gif . We easily prove that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ36_HTML.gif
(3.6)
Using the Weyl theorem [22] of a compact perturbation, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ37_HTML.gif
(3.7)
Since the operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq97_HTML.gif is selfadjoint, the eigenvalues of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq98_HTML.gif are real. From the definition of the eigenvalues, we get that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ38_HTML.gif
(3.8)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq99_HTML.gif denotes the set of all eigenvalues of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq100_HTML.gif .

Definition 3.2.

The multiplicity of a zero of the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq101_HTML.gif is called the multiplicity of the corresponding eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq102_HTML.gif .

Theorem 3.3.

Under the condition (1.14), the operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq103_HTML.gif has a finite number of simple real eigenvalues.

Proof.

To prove the theorem, we have to show that the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq104_HTML.gif has a finite number of simple zeros.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq105_HTML.gif be one of the zeros of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq106_HTML.gif . Now we show that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ39_HTML.gif
(3.9)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq107_HTML.gif be the Jost solution of (1.12) that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ40_HTML.gif
(3.10)
Differentiating (3.10) with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq108_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ41_HTML.gif
(3.11)
Using (3.10) and (3.11), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ42_HTML.gif
(3.12)
or
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ43_HTML.gif
(3.13)
It follows from (3.13) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ44_HTML.gif
(3.14)

that is, all zeros of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq109_HTML.gif are simple.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq110_HTML.gif denote the infimum of distances between two neighboring zeros of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq111_HTML.gif . We show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq112_HTML.gif . Otherwise, we can take a sequence of zeros https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq113_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq114_HTML.gif of the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq115_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ45_HTML.gif
(3.15)
It follows from (2.4) that, for large https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq116_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ46_HTML.gif
(3.16)

holds, where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq117_HTML.gif .

From the equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ47_HTML.gif
(3.17)
we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_Equ48_HTML.gif
(3.18)

There is a contradiction comparing (3.16) and (3.18). So https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq118_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F306571/MediaObjects/13661_2010_Article_914_IEq119_HTML.gif function has only a finite number of zeros.

Authors’ Affiliations

(1)
Department of Mathematics, Ankara University

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© Elgiz Bairamov et al. 2010

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